In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.

在这一贡献中，提出了应力梯度理论的有限元实现。该实现依赖于根据一个三阶主张量值场变量（即所谓的广义位移场）对偏微分方程的控制集进行的重新表述。广义位移场的体积部分与经典位移场密切相关，而偏斜部分可以用微位移来解释。此外，相关的弱公式用广义位移场或（完整）应力张量的法线投影规定了边界条件。对应力梯度弹性的典型边值问题的详细研究表明了所提出的公式的适用性。特别是，基于贝塞尔函数导出的承受拉力和扭力的圆柱杆的解析解，验证了有限元实现。在张力和扭转情况下，与相应的应变梯度弹性解形成鲜明对比的结果是，尺寸效应越小，效果越明显。